Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lower Bounds for Kernel Density Estimation on Symmetric Spaces

Dena Marie Asta

Abstract

We prove that kernel density estimation on symmetric spaces of non-compact type, whose L2-risk was bounded above in previous work (Asta,2021), in fact achieves a minimax rate of convergence. With this result, the story for kernel density estimation on all symmetric spaces is completed. The idea in adapting the proof for Euclidean space is to suitably abstract vector space operations on Euclidean space to both actions of symmetric groups and reparametrizations of Helgason-Fourier transforms and to use the fact that the exponential map for symmetric spaces of non-compact type defines a diffeomorphism.

Lower Bounds for Kernel Density Estimation on Symmetric Spaces

Abstract

We prove that kernel density estimation on symmetric spaces of non-compact type, whose L2-risk was bounded above in previous work (Asta,2021), in fact achieves a minimax rate of convergence. With this result, the story for kernel density estimation on all symmetric spaces is completed. The idea in adapting the proof for Euclidean space is to suitably abstract vector space operations on Euclidean space to both actions of symmetric groups and reparametrizations of Helgason-Fourier transforms and to use the fact that the exponential map for symmetric spaces of non-compact type defines a diffeomorphism.
Paper Structure (6 sections, 6 theorems, 24 equations)

This paper contains 6 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Assouad's Lemma
  4. Symmetric Spaces
  5. Helgason-Fourier Analysis
  6. Main Proof

Key Result

Theorem 1

Consider the following data. Then there exists a constant $K>0$ such that $\inf_{\hat{f}^n}\mathbb{E}_f[(\hat{f}^n-f)^2]\geq Kn^{-2\alpha/2\alpha+\dim\,\mathbf{X}}$, where the infimum is taken over all estimators $\hat{f}^n$ of $f$ based on $n$ samples drawn independently from $f$.

Theorems & Definitions (13)

  • Theorem 1
  • Corollary 1
  • Lemma 1: Assouad's Lemma, van-der-Vaart-asymptotic-stats
  • Lemma 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 3
  • proof
  • ...and 3 more